A person travels equal distances with speeds of 4 km/h, 6 km/h and 8 km/h. He takes a total time of 32.5 minutes. Find the total distance travelled by him

Options

A. A)3 km
B. B)4.5 km
C. C)7.5 km
D. D)6 km

Answer

To solve the problem step by step, let's denote the equal distance traveled in each segment as \( d \).

### Step 1: Calculate the time taken for each segment
The time taken to travel a distance \( d \) at different speeds can be calculated using the formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]

1. **Time at 4 km/h**:
\[ t_1 = \frac{d}{4} \]

2. **Time at 6 km/h**:
\[ t_2 = \frac{d}{6} \]

3. **Time at 8 km/h**:
\[ t_3 = \frac{d}{8} \]

### Step 2: Write the total time equation
The total time taken for all three segments is given as 32.5 minutes. We need to convert this time into hours for consistency with the speed units:
\[ 32.5 \text{ minutes} = \frac{32.5}{60} \text{ hours} = \frac{13}{24} \text{ hours} \]

Now, we can write the equation for total time:
\[ t_1 + t_2 + t_3 = \frac{d}{4} + \frac{d}{6} + \frac{d}{8} = \frac{13}{24} \]

### Step 3: Find a common denominator
To combine the fractions, we need to find a common denominator. The least common multiple of 4, 6, and 8 is 24. We can rewrite each term:
\[ \frac{d}{4} = \frac{6d}{24}, \quad \frac{d}{6} = \frac{4d}{24}, \quad \frac{d}{8} = \frac{3d}{24} \]

### Step 4: Combine the fractions
Now, we can combine the fractions:
\[ \frac{6d}{24} + \frac{4d}{24} + \frac{3d}{24} = \frac{(6d + 4d + 3d)}{24} = \frac{13d}{24} \]

### Step 5: Set the equation equal to the total time
Now, we set the combined time equal to the total time:
\[ \frac{13d}{24} = \frac{13}{24} \]

### Step 6: Solve for \( d \)
To solve for \( d \), we can multiply both sides of the equation by 24:
\[ 13d = 13 \]
Now, divide both sides by 13:
\[ d = 1 \text{ km} \]

### Step 7: Calculate the total distance traveled
Since the person travels the same distance \( d \) three times, the total distance \( D \) is:
\[ D = 3d = 3 \times 1 = 3 \text{ km} \]

### Final Answer
The total distance traveled by the person is **3 km**.

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